The best possible quadratic refinement of Sendov's conjecture
Abstract
A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if β is one of those roots, then within one unit of β lies a root of the polynomial's derivative. If we define r(β) to be the greatest possible distance between β and the closest root of the derivative, then Sendov's conjecture claims that r(β) 1. In this paper, we assume (without loss of generality) that 0 β 1 and make the stronger conjecture that r(β) 1-(3/10)β(1-β). We prove this new conjecture for all polynomials of degree 2 or 3, for all real polynomials of degree 4, and for all polynomials of any degree as long as all their roots lie on a line or β is sufficiently close to 1.
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